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E, ABCD is a rectangular based pyramid - Scottish Highers Maths - Question 3 - 2019
Question 3
E, ABCD is a rectangular based pyramid. AB = p, AD = q and AE = r. (a) Express BE in terms of p and r. Point F divides BC in the ratio 3:1. (b) Express vector EF ... show full transcript
Worked Solution & Example Answer:E, ABCD is a rectangular based pyramid - Scottish Highers Maths - Question 3 - 2019
Step 1
Step 2
Express vector EF in terms of p, q and r.
Answer
To express vector EF, we need to consider point F, which divides BC in the ratio 3:1. The coordinates can be identified based on the ratios:
We express BC as:
BC = rac{3}{4} imes BC + rac{1}{4} imes BC = rac{3}{4}(C - B)
Let’s define point F in terms of p, q, and r:
F = rac{3}{4}B + rac{1}{4}C
Next, we represent EF as:
EF = F - E = rac{3}{4}B + rac{1}{4}C - E
Substituting values for E, B, and C gives:
EF = rac{3}{4} egin{pmatrix} p \ 0 \ 0 \\ rac{1}{4} egin{pmatrix} 0 \ q \ 0 \\ E = egin{pmatrix} 0 \ 0 \ r \\ \Rightarrow EF = rac{3}{4} egin{pmatrix} p \ p \ p \\ - egin{pmatrix} 0 \ 0 \ r \\ = rac{3p}{4} + rac{1q}{4} - r
Thus, the complete expression for vector EF becomes:
EF = egin{pmatrix} rac{3}{4}p \ rac{1}{4}q - r \ rac{3}{4}p - r \\